Comparing Integers: In this article, we shall talk about integers, their definition, its representation on a number line. We shall also discuss how to add and subtract positive and negative integers.

There are some basic rules of comparison of integers in Math; they are greater than \(\left( > \right),\) less than \(\left( < \right),\) or equal sign \(\left( = \right).\) Then, we shall discuss how to identify the integer that is greater than, smaller than, or equals to another integer.

Integer

The term “integer” was taken in Mathem

atics from the Latin word “integer”, which means intact or whole. So, integers are very much like whole numbers, but it also includes negative numbers among them.

Definition: An integer is a number that doesn’t have the decimal or the fractional part from the set of negative and positive numbers, including zero.

Examples: The integers are: \( – 5,0,1,5,8,97\) and \(3043.\)

Number Line

On a number line, integers are written at equal intervals. The integers are categorized as positive integers and negative integers. If you observe the number line shown below, you can see that the origin is \(0\) (zero), which stands in the middle. There are positive numbers on the right side, and on the left side, there are negative numbers.

So, this number line represents the integers which include positive and negative numbers. The integers extend endlessly in both directions, which is why the arrows are marked at both ends to show that the line is infinite.

If you see the above number line carefully, the numbers increase and decrease as the direction changes. This is because any number on the left side is always less than the number to its right. For example, \(3\) is less than \(4; – 5\) is less than \( – 2.\) Similarly, any number on the right side is always more than the number to its left. 

Example: \(5\) is greater than the number \(3; – 1\) is greater than \( – 4.\) In the same way, fractions and rational numbers also are shown on the number line.

Positive and Negative integers

You know that a set of integers, which is defined as \(Z,\) includes:

1. Positive Integers: The positive integer is positive if it is greater than zero.
Example: \(1,2,3…\)
2. Negative Integers: The negative integer is negative if it is less than zero.
Example: \( – 1, – 2, – 3…\)
3. Zero is defined as neither negative nor positive integer.
\(Z = \left\{{… – 7, – 6, – 5, – 4, – 3, – 2, – 1,0,1,2,3,…} \right\}\)

Positive and Negative Integers on Number Line

Positive Integers: When you add any two positive numbers, the result will always be positive. Thus, while adding positive numbers, the direction of the move will always be to the right side.

Example: Addition of \(4\) and \(4\left({4 + 4 = 8} \right)\)
So the first number is \(4,\) and the second number is also \(4;\) both are positive numbers.
Locate the number \(4\) on a number line, then move \(4\) steps to the right side to reach \(8.\)

Negative Integers: When you add two negative numbers, the result you get is always negative. Thus, when you add negative numbers, the direction of the move will always be on the left side.

Example: The addition of the negative numbers \( – 3\) and \( – 5\)

Subtraction of Integers on Number Line

Positive Integers: When you want to subtract two positive numbers, move on to the left as far as the value of the second number.

Example: Subtract \(5\) from \(4\)

Here the first number is \(4,\) and the second number is \(5;\) both are positive. So, mark the number \(4\) on a number line. Then move \(5\) steps to the left will give \(-1.\)

Negative Integers: When you want to subtract the two negative numbers, then move towards the right side as far as the value of the second number.

Example: Subtract \( – 4\) from \( – 2\)

First, locate \( – 2\) on the number line, later move \( 4\) steps to the right to reach \( 2.\)

Greater Than and Less Than

Comparison tells us the similar properties of various objects. The primary concept in mathematics helps us describe whether the integers are similar, or one is greater than or one is smaller than the other, in comparing two integers.

We have mainly three special symbols that are used for comparing the integers. The basic symbols used in the comparison of integers are given below:

1. Greater than \(\left( > \right)\)
2. Less than \(\left( < \right)\)
3. Equals to \(\left( = \right)\)

Using the above-shown symbols, we can compare two integers of any type, such as natural numbers, whole numbers, integers and decimal numbers. Thus, comparing and exploring the differences between the integers is called the comparison of integers.

Zero Pairs

The zero pair is the pair of numbers whose sum is zero.
Example: \( + 1, – 1 = 0.\) used to illustrate addition and subtraction problems with the positive and the negative integers.

Compare and Order Integers

As you move to the right on the number line, integers get larger in value. As you move to the left on the number line, integers get smaller in value.

The rules of the ordering and the comparing of the integers are given below:

1. If we compare numbers with different signs, then the negative number is less than positive.
2. If numbers are both positive, then this is the case when we compare whole numbers.
3. If numbers are both negative, then we compare numbers without signs. The bigger is the positive number; the smaller is its corresponding negative number.

Solved Examples – Comparing Integers

Q.1. In a test, \(\left({ + 5} \right)\) marks are given for every correct answer, and \(\left({ -2} \right)\) are provided for every incorrect answer. Radhika answered all the questions and scored \(30\) marks though she got \(10\) correct answers.
Ans: Marks are given for one correct answer \( = 5\)
So, marks allotted for \(10\) correct answers \( = 5 \times 10 = 50\)
Radhika’s score \( = 30\)
Marks obtained for incorrect answers \( = 30 – 50 = \, – 20\)
Marks are given for one wrong answer \( = \left( { – 2} \right)\)
Hence, number of incorrect answers \( = \left({ – 20} \right) \div \left({ – 2} \right) = 10\)

Q.2. In a test, \(\left({ + 5} \right)\) marks are given for every correct answer and \(\left({ -2} \right)\) are provided for every incorrect answer. Jay answered all the questions and scored \(\left({ -12} \right)\) marks though he got 4 correct answers. How many wrong answers had they attempted?
Ans: Marks are given for one correct answer \( = 5\)
So, marks allotted for \(4\) correct answers \( = 5 \times 4 = 20\)
Jay scored \( = \left({ – 12} \right)\)
Marks obtained for incorrect answers \( = \left({ – 12} \right) – 20 = \, – 32\)
Marks are given for one wrong answer \( = \left({ – 2} \right)\)
Therefore, number of incorrect answers \( = \left({ – 32} \right) \div \left({ – 2} \right) = 16\)

Q.3. Which integers lie between \( – 8\) and \( – 2\)? Write the greater integer and smallest integer among them?

Ans: Integers between \( – 8\) and \( – 2\) are \( – 7, – 6, – 5, – 4, – 3.\)
The integer \( – 3\) is the largest, and \( – 7\) is the smallest integer.

Q.4. Compare the given integers \( + 9\) and \( -8.\) Show the greater integer between them by using the Mathematical symbol.
Ans: Given, \( + 9\) and \( -8\)
We can see that the digits \( 9\) has \( + \) sign and \( 8\) has a \( – \) sign. We know that an integer with a positive sign is always greater than an integer with a negative sign.
So, \( + 9\) is the greater integer.
In Mathematical form, it can be written as follows:
Hence, \( + 9 > – 8.\)

Q.5. Compare the given integers \( + 12\) and \( -14.\) Show the smaller integer among by using the Mathematical symbol.
Ans: Given, \( + 12\) and \( -14\)
We can see that the digits \( 12\) has \( + \) sign and \( 14\) has a \( – \) sign. We know that an integer with a negative sign is always smaller than an integer with a positive sign.
So, \( -14\) is the smaller integer.
In Mathematical form, it can be written as follows:
Hence, \( + 12 > – 14.\)

Summary

In the given article, we have discussed the definition of integers and then talked about the number line. Also, we covered how to add or subtract positive and negative integers on the number line, followed by greater than and less than. Finally, we discussed how to compare and order integers. Then you can see a few of the solved examples along with a few FAQs.

Frequently Asked Questions (FAQs)

Q.1. What are integers?
Ans: An integer is a number that doesn’t have the decimal or the fractional part from the negative and positive numbers set, including zero.

Q.2. How do you solve integers on a number line?
Ans: The example is provided below, showing how to subtract the integers on the number line. In the same way, you can solve the integers using different operations on the number line.
We will Subtract the numbers \(6\) from \(3.\)
Here the first number is \(3,\) and the second number is \(6;\) both are positive. So, mark the number \(3\) on a number line. Then move \(6\) steps to the left will give \(-3.\)

Q.3. How to identify which integer is greater?
Ans: By looking at the number line, we can easily tell which integers are greater (larger) than a certain number and which are less (smaller) than the number. When two numbers are located on a number line, the number to the right is larger than the number to the left.

Q.4. What are different symbols used to compare integers?
Ans: We have mainly three special symbols that are used for comparing the integers. The basic symbols used in the comparison of integers are given below:
1. Greater than \(\left( > \right)\)
2. Less than \(\left( < \right)\)
3. Equals to \(\left( = \right)\)

Q.5. Why is Z an integer symbol?
Ans: The notation \(Z\) for the set of the integers comes from the German word ‘Zahlen’, which means “numbers”. Integers are strictly larger than zero and are positive integers, and those that are less than zero are negative.

We hope this detailed article on comparing integers helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!